calculus .......

Question

Minimize the function  M=3x²+3ysubject to  2x+y=32 and find the values of x and y at which M is minimized.

The minimum value of M is ___, and it occurs when x is ___

and y is ___

Answer 1

Lagrange function L = 3x² + 3y - k(2x + y - 32);

L_x = 6x - 2k = 0; L_y = 3 - k = 0; L_k = - 2x - y + 32 = 0; k = 3; 6x - 6 = 0; x = 1; y = 30

M(1, 30) = 3 + 90 = 93. Le t x = 5 , y = 22 for example, then L(5, 22) = 75 + 66 = 141 > 93

So, M_min = 93 at x = 1, y = 30